Waves are naturally found in plasmas and have to be dealt with. This includes instabilities, fluctuations, waveinduced

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1 Lecure 1 Inroducion Why is i imporan o sudy waves in plasma? Waves are naurally found in plasmas and have o be deal wih. This includes insabiliies, flucuaions, waveinduced ranspor... Waves can deliver and/or manipulae energy-momenum in plasma: heaing, curren drive, paricle acceleraion mode sabilizaion, alpha channeling Raman and Brillouin scaering... Plasma diagnosics: inerferomery, Faraday roaion, Thomson scaering... Sudying plasma waves provides insigh ino wave physics in general. Basic equaions Maxwell s equaions A characerisic feaure of plasmas is ha hey conain free charges. Thus, any wave in a plasma is accompanied by oscillaing elecric currens, which unavoidably cause oscillaions of elecromagneic fields. Because of his, waves in plasmas are sudied using Maxwell s equaions. Overall, Maxwell s equaions include four equaions (or eigh equaions, if one couns componens independenly). However, Gauss s laws for elecric and magneic fields, aken ogeher wih charge conservaion, can be considered simply as iniial condiions for Ampere s and Faraday s laws. Thus, i will be enough for us o limi our consideraion o he laer wo, B 4 Π E 1 c c j 1 c B. E, A combinaion of hese equaions gives E 1 c or, equivalenly, 4 Π c j 1 c E 1 2 E c 2 4 Π 2 c 2 E j., Absen he curren densiy on he righ-hand side, his equaion describes ligh propagaion in vacuum. Thus, all feaures specific o plasma are, in fac, conained in he righ-hand side, i.e., he curren densiy. In general,

2 2 lecure-01.nb consiss of he self-consisen curren j ind induced by he field and i can also include an exernal curren j 0 independen of he wave field: j j ind j 0. Below, we consider j 0 prescribed and discuss how o model j ind. General linear media Linear media are defined as hose where j ind scales linearly wih he field E. In some cases, he relaion beween he wo can be as simple as j ind, x Σ, xe, x. This model is commonly known as Ohm s law. I is also called he local-response model, because i assumes ha he curren a, x is deermined only by he field a he same locaion, x. Tha can be a valid assumpion indeed, paricularly in srongly collisional plasma. Bu in general, he curren can be a nonlocal funcion of he field, i.e., exhibi dispersion. Nonlocaliy in ime is called emporal dispersion, and nonlocaliy in space is called spaial dispersion. The mos general expression for a dispersive linear curren can be wrien as follows. Suppose he dynamics is considered a 0, where 0 is some iniial momen of ime. Then, he curren induced by he field since ha ime can be wrien as j ind, x 0 ' 3 x' Σ, x, ', x'e ', x'. Commen: In lieraure, 0 is ofen aken o be o, bu he inegral does no always converge in his limi. Some subleies resuling from his fac will be discussed in his course laer. The ensor Σ, x, ', x', which is called (he ime-space represenaion of) conduciviy, can be defined formally as a funcional derivaive Σ j, x, x, ', x' E ', x' and serves as he weigh funcion ha deermines how much he field E ', x' conribues o he curren j ind, x. The fac ha he inegraion domain is limied o ' reflecs causaliy; namely, he curren a a given ime can be affeced by fields ha exised in he pas bu no in he fuure. In oher words, Σ, x, ', x' 0 for all '. Using j j 0 j ind and subsiuing he above expression for j ind, one ges he field equaion in he following inegro-differenial form: E 1 2 E c 2 4 Π j 0 4 Π 2 c 2 c 2 3 x' ' Σ, x, ', x'e ', x'. For he fuure references, le us represen his equaion also in wo oher forms as follows. Firs of all, one can wrie E 1 2 c 2 E, x ' 3 x' Χ, x, ', x'e ', x' 4 Π j 0 2 c 2, where Χ is called suscepibiliy (in he ime-space represenaion) defined via 0 ' 3 x' Χ, x, ', x'e ', x' 4 Π 0 ' 3 x' Σ, x, ', x'e ', x'. I is also convenien o inroduce he dielecric ensor (in he ime-space represenaion) as

3 lecure-01.nb 3 Ε, x, ', x' ' 0 x x' Χ, x, ', x', so he equaion for E (henceforh called wave equaion for breviy) becomes E 1 2 c 2 ' 3 x' Ε, x, ', x'e ', x' 4 Π j c 2. The added 0 in he ime argumen of he dela funcion denoes an infiniesimal posiive value. Adding his shif ensures ha he ime inegral over he domain 0, is well defined, because 0 ' 0 ' 0 0 Τ 0 Τ 0 Τ 0 Τ Τ Τ 1, whereas 0 Τ Τ is, sricly speaking, undefined. Saionary homogeneous linear media For now, le us limi our consideraion o saionary homogeneous linear media (SHLM). In such media, Σ is invarian wih respec o ime-space ranslaions, so i is allowed o depend only on he difference of ime-space coordinaes, x and ', x' raher han on heir absolue values. In oher words, one can express Σ as Σ, x, ', x' Σ ', x x'. Here, Σ is some ensor funcion, which we will also call conduciviy for breviy. Then, j ind, x 0 ' 3 x' Σ ', x x'e ', x'. Noe ha such Σ can be convenienly inerpreed as a Green s funcion. In oher words, Σ m,n, x can be undersood as he mh componen of he curren, j m, produced by he nh componen of he elecric field, E n, of he form E n ', x' ' 0 x'. The upper limi in he ime inegral can be replaced wih, since Σ is idenically zero a ' in any case. Le us also consider 0. Then, j is simply he convoluion of Σ and E, j ind, x ' 3 x' Σ ', x x'e ', x'. This expression is simplified significanly in he Fourier represenaion, j ind Ω, k ΣΩ, keω, k, so he specral represenaion of he conduciviy, ΣΩ, k, can be undersood as ΣΩ, k jω, k EΩ, k. (Noe ha his expression involves a regular parial derivaive, as opposed o a funcional derivaive ha emerges in he ime-space represenaion.) Explicily, i can be wrien as follows: ΣΩ, k Τ 3 Ω Τ ky y ΣΤ, y 0 Τ 3 y ΣΤ, y Ω Τ ky. The specral represenaions of he corresponding ensors Χ and Ε are as follows: ΧΩ, k 4 Π Ω ΣΩ, k,

4 4 lecure-01.nb ΕΩ, k 1 ΧΩ, k 1 4 Π ΣΩ, k. Ω Then, he field equaion becomes kkeω, k Ω2 or, equivalenly, kkeω, k k 2 EΩ, k Ω2 4 Π Ω ΕΩ, keω, k 2 c c 2 j 0 Ω, k, 4 Π Ω ΕΩ, keω, k 2 c c 2 j 0 Ω, k, Unlike he equaion in he ime-space represenaion, his equaion is no differenial, so i can be sudied easily as follows. Waves wih real frequencies Dispersion relaions Le us consider free oscillaions, which correspond o j 0 0. (A more careful approach will be discussed on he nex lecure.) Then he field equaion can be expressed as i, j Ω, k E j 0, where we inroduced he following marix: i, j c2 Ω 2 k i k j k 2 i, j Ε i, j Ω, k. Equaion (1) saes ha he field specrum in Ω, k space can be nonrivial only on surfaces (or, in onedimensional problems, curves) saisfying de Ω, k 0. This can be considered as an equaion for Ωk, called dispersion relaion. Depending on he number of soluions of he dispersion relaion, called branches, he corresponding waves are aribued as scalar waves ( 1) or vecor waves ( 1). Commen: For example, a ypical second-order (in ) wave equaion for a scalar field (e.g., he Klein-Gordon equaion) acually describes vecor waves, because i allows for boh posiive and negaive frequencies, i.e., corresponds o 2. The oher definiion commonly used in lieraure, scalar waves are hose described by scalar funcions, is ambiguous. For now, le us assume ha all hese soluions are real. Hence he general field can be represened as EΩ, k Ω Ω b k 2 Π E b k, b where E b k are deermined by iniial condiions. Noe ha each of hese mus saisfy vecor equaions Ω b k, ke b k 0. One can hus express he field specral funcion as E b b e b, where b k is some scalar ampliude, and e b is a polarizaion vecor defined via Ω b k, ke b k 0, e b 1. The field in he, x represenaion is obained via he inverse Fourier ransform, (1) (2)

5 lecure-01.nb 5 E, x b 3 k Ω 2 Π 3 2 Π Ω Ω bk 2 Π E b k Ω kx E b, x, E b, x E b k Ω bk kx 3 k 2 Π. 3 In oher words, he field can be decomposed ino independen conribuions from differen branches of he dispersion relaion. If only one branch is excied, a vecor wave hen behaves exacly as a scalar wave. Commen: The laer saemen is rue only in SHLM, whereas in nonsaionary or inhomogeneous medium, differen branches can inerac. Such effecs will be discussed laer in his course. Envelope evoluion. Group and phase velociies Suppose ha only one branch is excied, so Ωk kx E, x Ek 3 k 2 Π, 3 where we omied he branch index b for breviy. Suppose also ha Ek is localized around some k k 0, so we can expand Ωk as Ω Ω 0 v g k, where Ω 0 Ωk 0, k k k 0, and v g k Ωk k k0. This leads o E, x k 0 xv p Ek 0 k kxv g 3 k 2 Π 3, where we inroduced v p k 0 k 0 Ω k 0. Equivalenly, his can be wrien as E, x k 0 xv p Ex v g, where we inroduced Ex Ek 0 k k x 3 k 2 Π 3. In paricular, his implies E0, x k 0x Ex, so Ex represens he iniial envelope. Since, in he expression for E, x, he funcion E depends on, x only hrough he combinaion x v g, he envelope urns ou o be saionary in he frame moving wih velociy v g. Then, in he laboraory frame, he envelope ravels wih velociy v g ; i.e., v g is he velociy of envelope propagaion. I is also called he group velociy. Commen: Laer, we will explicily show ha, for linear waves considered here, v g is also he velociy of he wave acion ( phoons ), energy, and momenum, as well as he velociy of informaion. In conras, nonlinear waves can have heir acion, energy, momenum, and informaion propagaing a differen velociies, so he b

6 6 lecure-01.nb group velociy mus be defined in a more suble manner. These issues are beyond he scope of he course, which will be resriced o linear waves. Similarly, he phase facor in he expression for E, x depends on, x only hrough he combinaion x v p. Thus, v p serves as he velociy of phase frons. I is also called phase velociy. The phase and group velociies can be very differen (e.g., in some cases hey are even direced opposiely o each oher). The geomerical meaning of v g and v p in 1D is also seen in he following figure illusraing he propagaion of a localized pulse (ligher regions correspond o larger E 2 ): I is seen now ha knowing he dispersion relaions Ωk gives informaion no only abou monochromaic oscillaions bu also abou he evoluion of quasimonochromaic waves. (We will also exend his saemen o weakly inhomogeneous and nonsaionary plasmas laer.) In his sense, dispersion relaions deermine all wave physics. Because of his, dispersion relaions will be he primary (albei no he only) focus of our course.